Properties

Label 855.1.g.c
Level $855$
Weight $1$
Character orbit 855.g
Self dual yes
Analytic conductor $0.427$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -95
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 855.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.426700585801\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.475.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.347236875.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + q^{4} + q^{5} +O(q^{10})\) \( q -\beta q^{2} + q^{4} + q^{5} -\beta q^{10} + \beta q^{13} - q^{16} - q^{19} + q^{20} + q^{25} -2 q^{26} + \beta q^{32} -\beta q^{37} + \beta q^{38} + q^{49} -\beta q^{50} + \beta q^{52} + \beta q^{53} - q^{64} + \beta q^{65} + \beta q^{67} + 2 q^{74} - q^{76} - q^{80} - q^{95} -\beta q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{16} - 2 q^{19} + 2 q^{20} + 2 q^{25} - 4 q^{26} + 2 q^{49} - 2 q^{64} + 4 q^{74} - 2 q^{76} - 2 q^{80} - 2 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.41421
−1.41421
−1.41421 0 1.00000 1.00000 0 0 0 0 −1.41421
379.2 1.41421 0 1.00000 1.00000 0 0 0 0 1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.1.g.c 2
3.b odd 2 1 95.1.d.b 2
5.b even 2 1 inner 855.1.g.c 2
12.b even 2 1 1520.1.m.b 2
15.d odd 2 1 95.1.d.b 2
15.e even 4 2 475.1.c.b 2
19.b odd 2 1 inner 855.1.g.c 2
57.d even 2 1 95.1.d.b 2
57.f even 6 2 1805.1.h.b 4
57.h odd 6 2 1805.1.h.b 4
57.j even 18 6 1805.1.o.b 12
57.l odd 18 6 1805.1.o.b 12
60.h even 2 1 1520.1.m.b 2
95.d odd 2 1 CM 855.1.g.c 2
228.b odd 2 1 1520.1.m.b 2
285.b even 2 1 95.1.d.b 2
285.j odd 4 2 475.1.c.b 2
285.n odd 6 2 1805.1.h.b 4
285.q even 6 2 1805.1.h.b 4
285.bd odd 18 6 1805.1.o.b 12
285.bf even 18 6 1805.1.o.b 12
1140.p odd 2 1 1520.1.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 3.b odd 2 1
95.1.d.b 2 15.d odd 2 1
95.1.d.b 2 57.d even 2 1
95.1.d.b 2 285.b even 2 1
475.1.c.b 2 15.e even 4 2
475.1.c.b 2 285.j odd 4 2
855.1.g.c 2 1.a even 1 1 trivial
855.1.g.c 2 5.b even 2 1 inner
855.1.g.c 2 19.b odd 2 1 inner
855.1.g.c 2 95.d odd 2 1 CM
1520.1.m.b 2 12.b even 2 1
1520.1.m.b 2 60.h even 2 1
1520.1.m.b 2 228.b odd 2 1
1520.1.m.b 2 1140.p odd 2 1
1805.1.h.b 4 57.f even 6 2
1805.1.h.b 4 57.h odd 6 2
1805.1.h.b 4 285.n odd 6 2
1805.1.h.b 4 285.q even 6 2
1805.1.o.b 12 57.j even 18 6
1805.1.o.b 12 57.l odd 18 6
1805.1.o.b 12 285.bd odd 18 6
1805.1.o.b 12 285.bf even 18 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(855, [\chi])\):

\( T_{2}^{2} - 2 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( -2 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( -2 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( -2 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( -2 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( -2 + T^{2} \)
show more
show less